Theorem brcogw 4848

Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Assertion

Ref	Expression
brcogw	|- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A ( C o. D ) B )

Proof of Theorem brcogw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1003	. 2 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A e. V )
2		simpl2 1004	. 2 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> B e. W )
3		breq2 4049	. . . . . 6 |- ( x = X -> ( A D x <-> A D X ) )
4		breq1 4048	. . . . . 6 |- ( x = X -> ( x C B <-> X C B ) )
5	3, 4	anbi12d 473	. . . . 5 |- ( x = X -> ( ( A D x /\ x C B ) <-> ( A D X /\ X C B ) ) )
6	5	spcegv 2861	. . . 4 |- ( X e. Z -> ( ( A D X /\ X C B ) -> E. x ( A D x /\ x C B ) ) )
7	6	imp 124	. . 3 |- ( ( X e. Z /\ ( A D X /\ X C B ) ) -> E. x ( A D x /\ x C B ) )
8	7	3ad2antl3 1164	. 2 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> E. x ( A D x /\ x C B ) )
9		brcog 4846	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) )
10	9	biimpar 297	. 2 |- ( ( ( A e. V /\ B e. W ) /\ E. x ( A D x /\ x C B ) ) -> A ( C o. D ) B )
11	1, 2, 8, 10	syl21anc 1249	1 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A ( C o. D ) B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   E.wex 1515   e. wcel 2176   class class class wbr 4045   o. ccom 4680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-co 4685

This theorem is referenced by: (None)